How to maintain right angles while editing a rotated mesh?
This is happening because you're moving the loop according the its locals, which is not straight because your mesh is not straight--your mesh has its own locals (either an element that you added is making it this way, or you didn't make your mesh on the grid properly)--not according to the other points in respect to the loop. How do we know this? Because if you slide the loop G x 2, you find that this doesn't happen.

In orthographic view, you can see that there's an alignment issue of some sort at the very top (middle). There's a crease there to be specific (not easy to notice, I know). Check this by going into orthographic view from the front side (Numpad 5 for orthographic, and Numpad 1 for front) and look by the Z axis. Alternatively, look at Z-axis values of your edges.



The sign of not being straight (only noticeable at distances, generally): 
(Note that if the lines were straight, this should never happen.)